How to prevent timeline position moving when I click stop?
Edit:
Press Esc to stop the animation on the frame you started from.
Old:
As @moonboots suggests, you can easly set the starting point by pressing S before playing the animation.

You can use an alternative frame range.
You can switch between the main frame range and the temporary one by pressing the clock icon.

Once you set the starting point, you can jump to the first frame (Shift + Ctrl + Down Arrow) without pausing the animation.